integral of 1/(1+3cos^2(x))

解答

\int \frac{1}{1 + 3 cos ^{2} ( x )} d x

\frac{1}{2} arctan (\frac{tan ( x )}{2}) + C

求解步骤

\int \frac{1}{1 + 3 cos ^{2} ( x )} d x

使用三角换元法

= \int \frac{1}{u ^{2} + 4} d u

使用换元积分法

= \int \frac{1}{2 ( v ^{2} + 1 )} d v

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} arctan (v)

代回

= \frac{1}{2} arctan (\frac{tan ( x )}{2})

解答补常数

= \frac{1}{2} arctan (\frac{tan ( x )}{2}) + C

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